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$f: \mathbb{R^2\setminus\{0\}} \to \mathbb{R^2}$

$$f(x,y) = \left(\frac{x^2 - y^2}{x^2 + y^2}, \frac{xy}{x^2 + y^2}\right)$$

Is there a clean/nifty way to find the rank of the Jacobian of $f$ for any $p \neq 0$ and also describe the image of the function?

Currently, I've tried to find the partials and populate the Jacobian matrix but its turning out to be unwieldy. Is there a theorem that I should be aware of?

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    I propose you to change the cartesian coordinates to polar ($x=r\cos{\theta}, y=r\sin{\theta}$) and then return to cartesian coordinates: $$f(r,\theta) = (\cos{2\theta}, \sin{2\theta})$$ – 2017-02-14

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